Properties

Label 4000.1.bo.c
Level 4000
Weight 1
Character orbit 4000.bo
Analytic conductor 1.996
Analytic rank 0
Dimension 8
Projective image \(D_{20}\)
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 4000.bo (of order \(20\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{20}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( -\zeta_{20}^{9} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{20}^{9} q^{9} \) \( + ( \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{13} \) \( + ( \zeta_{20}^{2} + \zeta_{20}^{9} ) q^{17} \) \( + ( \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{29} \) \( + ( 1 + \zeta_{20}^{3} ) q^{37} \) \( + ( -\zeta_{20} - \zeta_{20}^{7} ) q^{41} \) \( + \zeta_{20}^{5} q^{49} \) \( + ( 1 + \zeta_{20}^{9} ) q^{53} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{61} \) \( + ( -\zeta_{20}^{3} + \zeta_{20}^{4} ) q^{73} \) \( -\zeta_{20}^{8} q^{81} \) \( + ( 1 + \zeta_{20}^{2} ) q^{89} \) \( + ( -\zeta_{20} + \zeta_{20}^{8} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{7}\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 0.809017i
0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0 0 0 0 0 0 0 −0.951057 + 0.309017i 0
993.1 0 0 0 0 0 0 0 0.951057 0.309017i 0
1793.1 0 0 0 0 0 0 0 0.587785 + 0.809017i 0
1857.1 0 0 0 0 0 0 0 0.951057 + 0.309017i 0
2593.1 0 0 0 0 0 0 0 −0.587785 + 0.809017i 0
2657.1 0 0 0 0 0 0 0 0.587785 0.809017i 0
3393.1 0 0 0 0 0 0 0 −0.951057 0.309017i 0
3457.1 0 0 0 0 0 0 0 −0.587785 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3457.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
25.f Odd 1 yes
100.l Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{13}^{8} - \cdots\) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).